图上的表示学习
定义
图 （也被称为简单图、无向图） 可以表示为 \(\mathcal{G} = \{\mathcal{V}, \mathcal{E}\}\)。
\(\mathcal{V}\) 是 顶点 集(也可以称为 节点 或者 点);
\(\mathcal{E} \subseteq \{ \{x, y\} \mid x, y \in \mathcal{V}~and~x \neq y \}\) 是 边 集(也可以称为 连接 或者 线), 其为顶点间的 无序对 (也就是说，一条边与两个不同的顶点相关联)。
在边 \(\{x, y\}\) 中, 顶点 \(x\) 和 \(y\) 被称为边的 端点。 称这条边 连接 \(x\) 和 \(y\) 并且 关联 \(x\) 和 \(y\)。 可能存在孤立顶点。 两条或更多的边连接同样的顶点称为 重边 ，根据以上定义未被允许。
结构构建
图的关联结构可以通过以下方法构建。详细参考 这里。
边列表 (默认)
dhg.Graph从超图关联结构简化而来
在如下的例子中，我们随机生成一个图关联结构和一个特征矩阵，并对此结构进行一些基本的学习操作。
>>> import torch >>> import dhg >>> # Generate a random graph with 5 vertices and 8 edges >>> g = dhg.random.graph_Gnm(5, 8) >>> # Generate a vertex feature matrix with size 5x2 >>> X = torch.rand(5, 2) >>> # Print information about the graph and feature >>> g Graph(num_v=5, num_e=8) >>> # Print edges in the graph >>> g.e[0] [(0, 1), (2, 4), (0, 4), (3, 4), (0, 3), (2, 3), (0, 2), (1, 3)] >>> # Print the both side of edges in the graph >>> g.e_both_side[0] [(0, 1), (2, 4), (0, 4), (3, 4), (0, 3), (2, 3), (0, 2), (1, 3), (1, 0), (4, 2), (4, 0), (4, 3), (3, 0), (3, 2), (2, 0), (3, 1)] >>> # Print vertex features >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]])
基于谱域的学习
使用GCN的拉普拉斯进行平滑
>>> # Print the Laplacian matrix defined by GCN that associated with the graph structure >>> g.L_GCN.to_dense() tensor([[0.2000, 0.2582, 0.2236, 0.2000, 0.2236], [0.2582, 0.3333, 0.0000, 0.2582, 0.0000], [0.2236, 0.0000, 0.2500, 0.2236, 0.2500], [0.2000, 0.2582, 0.2236, 0.2000, 0.2236], [0.2236, 0.0000, 0.2500, 0.2236, 0.2500]]) >>> # Print the vertex features befor feautre smoothing >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> X_ = g.smoothing_with_GCN(X) >>> # Print the vertex features after GCN-based smoothing >>> X_ tensor([[0.5434, 0.6609], [0.5600, 0.5668], [0.3885, 0.6289], [0.5434, 0.6609], [0.3885, 0.6289]])
使用对称归一化的拉普拉斯进行平滑
>>> # Print the symmetrically normalized Laplacian matrix associated with the graph structure >>> g.L_sym.to_dense() tensor([[ 1.0000, -0.3536, -0.2887, -0.2500, -0.2887], [-0.3536, 1.0000, 0.0000, -0.3536, 0.0000], [-0.2887, 0.0000, 1.0000, -0.2887, -0.3333], [-0.2500, -0.3536, -0.2887, 1.0000, -0.2887], [-0.2887, 0.0000, -0.3333, -0.2887, 1.0000]]) >>> # Print the vertex features >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> X_ = g.smoothing(X, g.L_sym, 0.1) >>> # print the new vertex features >>> X_ tensor([[ 0.3746, 0.9525], [ 0.7926, 0.3590], [-0.0210, 0.3251], [ 0.8218, 0.7940], [ 0.4756, 0.6351]])
使用左归一化（随机游走）的拉普拉斯进行平滑
>>> # Print the left(random-walk) normalized Laplacian matrix associated with the graph structure >>> g.L_rw.to_dense() tensor([[ 1.0000, -0.2500, -0.2500, -0.2500, -0.2500], [-0.5000, 1.0000, 0.0000, -0.5000, 0.0000], [-0.3333, 0.0000, 1.0000, -0.3333, -0.3333], [-0.2500, -0.2500, -0.2500, 1.0000, -0.2500], [-0.3333, 0.0000, -0.3333, -0.3333, 1.0000]]) >>> # Print the vertex features >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> X_ = g.smoothing(X, g.L_rw, 0.1) >>> # Print the new vertex features >>> X_ tensor([[ 0.3843, 0.9603], [ 0.7752, 0.3341], [-0.0263, 0.3174], [ 0.8316, 0.8018], [ 0.4703, 0.6275]])
基于空域的学习
从顶点到顶点的消息传递
>>> # Print the vertex messages >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> X_ = g.v2v(X, aggr="mean") >>> # Print the new vertex messages >>> X_ tensor([[0.5107, 0.5386], [0.5946, 0.8515], [0.5512, 0.7786], [0.4113, 0.5738], [0.4051, 0.6875]])
从顶点到顶点依赖边权的消息传递
>>> # Print the vertex messages >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> g.e_weight tensor([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) >>> # Generate random edge weights >>> e_weight = torch.rand(len(g.e_weight)) >>> e_weight tensor([0.6689, 0.2302, 0.8003, 0.7353, 0.7477, 0.5585, 0.6226, 0.8429, 0.6105, 0.1248, 0.8265, 0.2117, 0.8574, 0.4282, 0.3964, 0.1440]) >>> X_ = g.v2v(X, e_weight=e_weight, aggr="softmax_then_sum") >>> # Print the new vertex messages >>> X_ tensor([[0.5648, 0.5657], [0.5758, 0.8582], [0.5699, 0.7794], [0.4720, 0.5493], [0.3742, 0.6827]])